4.5 r-Modes

r-modes are quasitoroidal oscillations in rotating fluids that occur because of the Coriolis effect (akin to the Rossby modes studied in planetary “spots”). In GWs, these modes are driven unstable by gravitational radiation reaction. The primary questions surrounding these instabilities arise in calculating the saturation amplitude of r-modes. The GW emission from r-mode unstable neutron-star remnants of core-collapse SNe would be easily detectable if αmax ∼ 1. In rotating stars, gravitational radiation reaction drives the r-modes toward unstable growth [4, 96]. In hot, rapidly-rotating neutron stars, this instability may not be suppressed by internal dissipative mechanisms (such as viscosity and magnetic fields) [179]. If not limited, the dimensionless amplitude α of the dominant (m = 2) r-mode will grow to order unity within ten minutes of the formation of a neutron star rotating with a millisecond period. The emitted GWs carry away angular momentum, and will cause the newly-formed neutron star to spin down over time. The spin-down timescale and the strength of the GWs themselves are directly dependent on the maximum value αmax to which the amplitude is allowed to grow [180Jump To The Next Citation Point, 181Jump To The Next Citation Point].

Originally, it was thought that αmax ∼ 1. In such a case, we can estimate the GW signal from stellar collapse. We expect multiple GW bursts to occur as material falls back onto the neutron star and results in repeat episodes of r-mode growth (note that a single r-mode episode can have multiple amplitude peaks [180Jump To The Next Citation Point]). Using Equation (10View Equation), FHH [106Jump To The Next Citation Point] calculate that the characteristic amplitude of the GW emission from this r-mode evolution tracks from 6 – 1 × 10–22, over a frequency range of 103 – 102 Hz for a source driven by fallback at 10 Mpc. They estimate the emitted energy to exceed 1052 erg.

Later work indicated that αmax may be ≥ 3 [180, 294, 268, 181Jump To The Next Citation Point]. But most recent research suggests that magnetic fields, hyperon cooling, and hyperon bulk viscosity may limit the growth of the r-mode instability, even in nascent neutron stars [159, 158, 248, 249, 181, 178, 132, 5] (significant uncertainties remain regarding the efficacy of these dissipative mechanisms). One way to reduce this viscosity is to invoke non-standard physics in the dense equation of state, e.g., quark material, anti-kaon core [53, 71]. Even with this more exotic physics, the reduction in viscosity is limited to specific regions in the neutron star and in the spin/temperature phase space. More work is needed to determine if such modifications can allow r-modes to make a detectable signal.

In addition, a study of a simple barotropic neutron star model by Arras et al. [8] argue that multimode couplings could limit αmax to values ≪ 1. If αmax is indeed ≪ 1 (see also [125, 27, 26, 28, 252, 24]), GW emission from r-modes in collapsed remnants is likely undetectable. From Equation (10View Equation), we can see that the GW signal is proportional to the mode amplitude, so a decrease in the maximum amplitude by an order of magnitude corresponds to an order of magnitude decrease in the GW strain. If correct, and much of the community believes the maximum of the mode amplitude may be even smaller than 0.1, the GW signal from r-modes is much lower than any other GW source and will not contribute significantly to the observed signal in stellar collapse. Because of this, r-mode sources are omitted from figures comparing source strengths and detector sensitivities and from discussions of likely detectable sources in the concluding section.

r-modes may still produce signals in accreting systems such as low-mass X-ray binaries. Work continues in this subject for these systems, but this is beyond the subject of this review.


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